reserve X for set;
reserve a,b,c,k,m,n for Nat;
reserve i,j for Integer;
reserve r,s for Real;
reserve p,p1,p2,p3 for Prime;
reserve z for Complex;

theorem
  for D being Integer holds
  { [x,y,z] where x,y,z is positive Nat : x^2-D*y^2 = z^2 } is infinite
  proof
    let D be Integer;
    D < 0 or D = 0 or D > 0;
    hence thesis by Lm52,Lm53,Lm54;
  end;
